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• Tags
  • #projects/notes/reading #geomtop
• Refs:
  • Extremely good review #resources/notes
  • https://youtu.be/vWFjqgb-ylQ #resources/videos
  • Video of conformal flows
  • Gradient Descent
  • https://arxiv.org/pdf/2005.10799.pdf#page=14
  • Hiro-Lurie https://www.math.ias.edu/~lurie/papers/Broken-new.pdf#page=1
• Links:
  • Unsorted/handle decomposition
  • Stein

Morse Theory

The standard procedure:

• Show that $D$ is a Fredholm operator
• Show that $D$ is surjective, so ${\mathsf{Ind}}D = \dim \ker D$
• Show a moduli space is the intersection of some section $s$ of a bundle with the zero section.
• Show that this intersection is transverse, i.e. $Ds$ is surjective.
  • Vary the Riemannian metric and use a second category theorem to get the Morse-Smale condition.
• Apply the infinite-dimensional inverse function theorem
• Show that a Frechet manifold is in fact a Banach manifold and apply a version of Sard’s Theorem

Motivations

• Can be used to prove the high dimensional case of the generalized Unsorted/Poincare conjectures

Results

Theorem: Every compact smooth manifold admits a Morse function.

Theorem: Morse function are generic: given any smooth function $f: X\to Y$, there is an arbitrarily small perturbation of $f$ that is Morse.

See Morse lemma

Theorem 3: If $(W; M_0, M_1) \to I$ is Morse with no critical points then $W \cong_{\operatorname{Diff}} I \times M_0$

Theorem: If $X$ is closed and admits a Morse function with exactly 2 critical points, $X$ is homeomorphic to $S^n$.

Possibly used in Milnor’s Unsorted/parahoric 7-sphere (show a diffeomorphism invariant differs but admits such a Morse function)

Theorem: $M$ is homotopy equivalent to a CW complex with one cell of dimension $k$ for each critical point of $f$ of index of a Morse function $k$.

Gradients

# Energy

Critical points

Idea: the number of linearly independent direction you can move for which the function decreases.

Morse chain complex

Morse inequalities

Broken trajectories

Moduli space of flow lines

Zero set of a section

See vertical and horizontal subspace

Sard

Pictures

Examples

Notes

Dave’s Videos

• Historic note: Morse wanted to know not the diffeomorphism type of $M$, but rather the homotopy type.

• Definition: critical values and critical points

• Definition: critical point

• Theorem (Smale, h-cobordism theorem

  • If $X^n$ is a smooth cobordism.

• Corollary (High-Dimensional Unsorted/Poincare conjectures

  • If $X_1^n, X_2^n \cong_{\operatorname{Diff}} S^n$, then there exists an h-cobordism between them.
  • Proof: use algebraic topology to eliminate (cancel) critical points.

• Definition: index of a Morse function

  • Look at coordinate-free def?
  • Standard form at critical points
  • Alternatively: Hessian is non-singular at every critical point.
  • $f^{-1}{{\partial}}Y) = {{\partial}}X$

• Definition: Stable and generic

• Definition: cobordism

  • Example: (pair of pants)
  • Category: Objects are manifolds, morphisms are cobordisms between them

• Consequence of theorem 3: $M_0 \cong_{\text{Diff}} M_1$ is a diffeomorphism, useful to show two things are diffeomorphic, used in higher-dimensional Poincare.

  • Recall that this is proved by constructing a vector field $V$ on $W$, then using a diffeomorphism $\phi:I \times M_0 \to W$ by flowing along $V$.
  • Can we do gradient flow in the presence of a metric? #todo/questions

Intro Video

https://www.youtube.com/watch?v=78OMJ8JKDqI

Morse theory: handles nice singularities. Can have worse ones, covered by [dynamical systems](catastrophe theory](dynamical%20systems](catastrophe%20theory) (dynamical systems).

Importance of CW complexes: triangulation of surfaces.

See Morse lemma

Morse Theorem 1: If there are no critical points, $M_A \simeq M_B$.

Stable vs unstable manifolds:

Consider height function on torus. Circles are index 0 critical points, triangle is index 1.

Cancellation:

Can use persistent homology to measure “importance” of critical points.

Unsorted

https://youtu.be/mIUi1zIUQJw?t=42

• Diffeomorphism type depends on isotopy classes of attaching maps.

See handle decomposition

More Notes

Historic note: Morse wanted to know not the diffeomorphism type of $M$, but rather the homotopy type. - Theorem (Smale, h-cobordism) - If $X^n$ is a smooth cobordism, $n\geq 6$, $\pi_1(X) = 0$, and $X$ “looks like” a product in algebraic topology, then $X$ is a product cobordism. - Corollary (High-Dim Poincare) - If $X_1^n, X_2^n \cong_{\operatorname{Diff}} S^n$, then there exists an $h{\hbox{-}}$cobordism between them. - Proof: use algebraic topology to eliminate (cancel) critical points. - Theorem: Every compact manifold has a Morse function. - Theorem: Morse functions are generic (given any smooth function $f: X\to Y$, there’s an arbitrarily small perturbation of $f$ that is Morse). - Theorem (Morse Lemma): If $p\in {\mathbf{R}}^n$ is a critical point of $f: {\mathbf{R}}^n \to {\mathbf{R}}$ such that the Hessian $H_f(p)$ is a non-degenerate bilinear form, then $f$ is locally Morse (standard form). - Theorem: If $(W; M_0, M_1) \to I$ is Morse with no critical points then $W \cong_{\operatorname{Diff}} I \times M_0$ - Consequence: $M_0 \cong_{\text{Diff}} M_1$ is a diffeomorphism, useful to show two things are diffeomorphic, used in higher-dimensional Poincare. - Theorem: If $X$ is closed and admits a Morse function with exactly 2 critical points, $X$ is homeomorphic to $S^n$. - Possibly used in Milnor’s exotic 7-sphere (show a diffeomorphism invariant differs but admits such a Morse function) - Diffeomorphism type depends on isotopy classes of attaching maps.

Morse Theory

Goal: handlebody decomposition, or for the purposes of the above theorems, retractions onto a CW complex. Decomposing a cobordism into a sequence of elementary cobordisms (admit a Morse function with a single critical point).

Fact: since $\phi$ is Morse, $M^{2n}$ can be retracted onto a complex of dimension $d\leq n$, since all critical points will have index $\leq n$.

Note: this immediately implies the Lefschetz Hyperplane theorem for affine manifolds $N$, i.e. that they are entirely determined by the homology and homotopy of $N\cap H$ for any hyperplane. Very strong!

Setting up notation/definitions:

• $V$ will be a smooth $n{\hbox{-}}$manifold
• $W$ an $n{\hbox{-}}$dimensional cobordism
• $\phi: V\to {\mathbf{R}}$ a smooth function
• $p$ a critical point of $\phi$ (i.e. the derivative $d_p \phi$ vanishes)
• $H_p\phi = ({{\partial}^2 \phi \over {\partial}x_i^2 {\partial}x_j^2})$ the Hessian matrix
• $\null_\phi(p)$ the nullity of $\phi$ at $p$ is $\dim \ker H_p$, regarding $H_p\phi$ as a symmetric bilinear form on $T_p V$
• $p$ is nondegenerate iff $\null_\phi(p) = 0$.
• The Morse index at $p$ is the dimension of the maximal subspace on which the associated quadratic form $H_p \phi$ is negative definite.

Theorem (Morse Lemma)

Near a nondegenerate critical point $p$ of $\phi$ of index $k$ there exists a smooth coordinate chart $U$ and coordinates $\mathbf{x} \in {\mathbf{R}}^n$ such that $\phi$ has the form $$\begin{align*}\phi(\mathbf{x}) = \phi(p) + \mathbf{x}^t A \mathbf{x}\end{align*}$$ where $A = \operatorname{diag}(-1, \cdots, -1, 1,\cdots 1)$.

Toward a generalization, we can also write ${\mathbf{R}}= {\mathbf{R}}^{k} \times{\mathbf{R}}^{n-k}$ and $$\begin{align*} \phi(\mathbf{x}_1, \mathbf{x}_2) = \phi(p) - {\left\lVert {\mathbf{x}_1} \right\rVert}^2 + {\left\lVert {\mathbf{x}_2} \right\rVert}^2 \end{align*}$$

Lemma (The nondegenerate directions can be split off)

If $\null_\phi(p) = \ell$ then we can instead write ${\mathbf{R}}= {\mathbf{R}}^{n-k-\ell} \times{\mathbf{R}}^k \times{\mathbf{R}}^\ell$ and $$\begin{align*} \phi(\mathbf{x}, \mathbf{y}, \mathbf{z}) = {\left\lVert {\mathbf{x}} \right\rVert}^2 - {\left\lVert {\mathbf{y}} \right\rVert}^2 + \psi(\mathbf{z}) \end{align*}$$ where $\psi: {\mathbf{R}}^\ell \to {\mathbf{R}}$ is some smooth function.

Definition

A degenerate critical point is embryonic iff $\null_\phi(p) = 1$ and writing $L = \ker H_p\phi = \mathop{\mathrm{span}}_{\mathbf{R}}{\mathbf{v}}$, the third directional derivative $D^3_{\mathbf{v}}\phi$ (?) is nonzero.

We now consider homotopies of Morse functions $\phi: I \times V \to {\mathbf{R}}$, where we can partially apply the $I$ factor to get a 1-parameter family $\left\{{\phi_t {~\mathrel{\Big\vert}~}t\in I}\right\}$.

Definition

A homotopy $\Phi: V\times I \to {\mathbf{R}}$ of Morse functions has a birth-death type critical point at $p$ at $t=t_0$ iff $p$ is embryonic for $\phi_0$ and $(t_0, p)$ is a nondegenerate critical point of $\Phi$.

Recall what a Cerf diagram/profile is – I don’t

Theorem (Whitney)

In three parts:

• Near an embryonic critical point $p$ of $\phi$ of index $k$ there exist coordinate $(\mathbf{x}, \mathbf{y}, z) \in {\mathbf{R}}^{n-k-1} \oplus {\mathbf{R}}^{k} \oplus {\mathbf{R}}$ such that $\phi$ has the form $$\begin{align*} \phi(\mathbf{x}, \mathbf{y}, z) = \phi(p) + {\left\lVert {\mathbf{x}} \right\rVert}^2 - {\left\lVert {\mathbf{y}} \right\rVert}^2 + z^3 \end{align*}$$

• If $p$ is birth-death type for $\Phi$ of index $k$, then up to conjugating $\phi_t$ by a (uniform in $t$) family of diffeomorphisms, each $\phi_t$ is of the form $$\begin{align*} \phi(\mathbf{x}, \mathbf{y}, z) = \phi(p) + {\left\lVert {\mathbf{x}} \right\rVert}^2 - {\left\lVert {\mathbf{y}} \right\rVert}^2 + z^3 \pm tz \end{align*}$$

• Any two homotopies $\Phi, \Phi'$ with points $(p, 0)$ and $(p', 0)$ with the same index and Cerf profile differ only by a diffeomorphism, i.e. there is a family of diffeomorphisms $h_t$ such that $\phi'_t \circ h_t = \phi_t$ for every $t$.

• A generic $\Phi$ has only nondegenerate and birth-death type singularities.

Definition

A singularity is birth type if the sign on $t$ is positive, and death type if negative.

Fact

Embryonic critical points are isolated, near a birth-type singularity two nondegenerate critical points of indices $k, k-1$ emerge, and near a death type they merge and disappear.

Pretty vague – I know there are pictures here that make this more obvious, but I couldn’t find much.

Definition

A cobordism is a triple $(W; M_+, M_-)$ where $W$ is an oriented compact smooth manifold with cooriented boundary ${{\partial}}W = M_+ {\textstyle\coprod}M_- = {{\partial}}_- W {\textstyle\coprod}{{\partial}}_+ W$, where the coorientation is provided by the inward (resp. outward) normal vector field (???). We’ll usually just denote this as $W$.

Definition

A Lyapunov cobordism is a triple $(W, \phi , X)$ where

• $W$ is a usual cobordism,
• $\phi: W\to {\mathbf{R}}$ is a smooth functional that is constant and has no critical points when restricted to ${{\partial}}W$,
• $X$ is a gradient-like vector field for $\phi$ which points inward along ${{\partial}}_- W$ and outward along ${{\partial}}_+ W$.

Definition

Such a cobordism is elementary iff there exist no $X{\hbox{-}}$trajectories between distinct critical points of $\phi$.

Theorem (Smale, h-cobordism)

Let $W$ be a cobordism of dimension $W\geq 6$ such that $W, {{\partial}}_{\pm}W$ are simply connected, and $H_*(W, {{\partial}}_- W; {\mathbf{Z}}) = 0$. Then $W$ admits a Morse function without critical points which is constant on ${{\partial}}_\pm W$.

In particular, $W \cong I\times M$ is diffeomorphic to the trivial product cobordism, and $M\cong N$ are diffeomorphic.

Proof (Sketch)

Goal: find a handle decomposition with no handles, then integrate along the gradient vector field of a Morse function $\phi$ to get a diffeomorphism.

• Find a Morse function and induce a handle decomposition
• Rearrange handles so that lower index handles are attached first
• Define a chain complex as free ${\mathbf{Z}}{\hbox{-}}$module on handles with boundary given in terms of intersection numbers of attaching spheres $k$ and belt $k-1$ spheres
• Find $k{\hbox{-}}$handles, create a pair of $k+1, k+2$ handles such that the $k+1$ handle cancels/fills in the $k{\hbox{-}}$handle (not sure why the $k+2$ is needed here)
• End up with nothing but an $n{\hbox{-}}$handle and an $n-1{\hbox{-}}$handle – turn “upside down” and repeat process with $-\phi$ to remove them.

Proof (Sketch)

• Pick $\phi: W\to {\mathbf{R}}$ Morse such that ${{\partial}}_\pm W$ are regular level sets.
• Make $\phi$ self-intersecting (uses a transversality argument)
• Partition manifold into regular level sets $L_k \coloneqq\phi^{-1}(k - \frac 1 2)$ for each $k\in {\mathbb{N}}$.
• Letting $\left\{{p_i}\right\}$ be the critical points in $L_k$ and $\left\{{q_j}\right\}$ the critical points in $L_{k-1}$, form the matrix $A$ of intersection numbers $S_{p_i}^- \smile S_{q_j}^+$ between the stable sphere of $p_i$ and the unstable sphere of $q_j$.
• Goal: since homology can be read off $SNF(A)$, and we know $H_* = 0$ here, we try to reduce $A$ to SNF with geometric operations
  • Handle slides: Add row $j$ to row $i$ by moving $p_i$ to $L_{k+j}, ~j\geq 1$, deform $X$ to produce a trajectory $p_j \to p_i$, then “the stable manifold of $p_i$ slides over the stable manifold of $p_j$” (?) replacing $[S_i^-]$ with $[S_i^-] + [S_j^-]$ in homology.
  • This makes $A = [I, 0; 0, 0]$ a block matrix with the identity in the top-left.
  • Handle Cancellation: Take two transverse intersection points $z_+, z_-$ with local intersection indices $1, -1$, connect via two paths: one in $S_i^-$, one in $S_j^+$. This yields a map $S^1 \hookrightarrow L_k$, use the Whitney trick to fill with an embedded disc $\Delta$, then push $S_i^-$ over $\Delta$ eliminates $z_\pm$.
  • This leaves a collection $S_i^-, S_i^+$ for $i=1,\cdots, r$ intersecting in a single point $z_0$, then (lemma) there are unique trajectories $q_i \to p_i$ for each $I$ and thus they can be eliminated.
• Do this in $L_k$; we now have a Morse function with no critical points except possibly of index $0, 1, n-1$, or $n$.
• Use “Smale’s trick”: trades in an index $k$ critical point for one of index $k+1$ and one of index $k+2$, such that $k, k+1$ cancel. Trade index $1$ for index $2, 3$ and cancel index $3$ as before.
• Eliminate $0, n$ with a lemma (unclear)